One interpretation of this phrase is that Jesus was making a veiled political statement to the Israelites.
In his culture, apparently the right hand was used for all dealings with others, and the left hand for wiping one's ass and other manipulations of a personal nature.
If you smacked someone across the face with the back of your hand, it meant that you considered that person to be sub-equal.
However, if you slapped someone with the palm of your hand, it meant that the two of you were considered to be on the same social plane.
The Israelites were used to being treated as less than equal by the Romans.
To turn the other cheek meant that if they were hit again, it would be with the flat of the hand. Thus they would be making a silent statement of insubordination to second-class status if they did so.
Jesus the revolutionary. How cool.